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Physics and Machine Learning: EmergingParadigms

Jose D. Martın-Guerrero1, Paulo J. G. Lisboa2, Alfredo Vellido3

1- Dept. of Electronic Engineering - Universitat de ValenciaAv. de la Universitat, s/n, 46100 Burjassot (Valencia) - Spain

2- Dept. of Mathematics and Statistics - Liverpool John Moores UniversityByrom St. L3 3AF Liverpool - United Kingdom

3- Dept. of Computer Science - Univ. Politecnica de CatalunyaC. Jordi Girona, 1-3, 08034 Barcelona - Spain

Abstract.

Current research in Machine Learning (ML) combines the study of varia-tions on well-established methods with cutting-edge breakthroughs basedon completely new approaches. Among the latter, emerging paradigmsfrom Physics have taken special relevance in recent years. Although stillin its initial stages, Quantum Machine Learning (QML) shows promisingways to speed up some of the costly ML calculations with a similar oreven better performance than existing approaches. Two additional ad-vantages are related to the intrinsic probabilistic approach of QML, sincequantum states are genuinely probabilistic, and to the capability of findingthe global optimum of a given cost function by means of adiabatic quan-tum optimization, thus circumventing the usual problem of local minima.Another Physics approach for ML comes from Statistical Physics and islinked to Information theory in supervised and semi-supervised learningframeworks. On the other hand, and from the perspective of Physics, MLcan provide solutions by extracting knowledge from huge amounts of data,as it is common in many experiments in the field, such as those related toHigh Energy Physics for elementary-particle research and ObservationalAstronomy.

1 Introduction

ML has become indispensable for pattern discovery from large data sets. It is alsothe theory at the core of a larger set of data science tools known as data mining.It is a mature field with an astonishing array of practical applications in tasksrelated to modeling, prediction, classification, visualization and planning [1].ML includes many robust methods that can transform raw data into structuredinformation by means of learning algorithms [2].

It is hence not surprising that those learning algorithms are starting to findtheir way into applications of quantum information processing in three differentways. First, using quantum resources to perform learning is an attractive line ofinquiry, with advantages ranging from quadratic or even exponential speedup,increased learning capacity, reduced sample complexity and better generalizationperformance; examples include quantum perceptrons, quantum neural networks,

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ESANN 2016 proceedings, European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning. Bruges (Belgium), 27-29 April 2016, i6doc.com publ., ISBN 978-287587027-8. Available from http://www.i6doc.com/en/.

quantum deep learning, boosting training by Grover’s search, adiabatic quan-tum annealing, and quantum support vector machines [3]. Second, from theperspective of quantum theory foundations, causal networks and their quantumgeneralizations, and the connection to non-locality are of paramount importance,whereas solid-state implementations of adiabatic quantum evolution are alreadybringing tangible benefits to ML practitioners [4]. Finally, classical ML appliedin experimental Physics is also proving to be a fruitful avenue that deservesfurther exploration; this is an applied approach with relevance to experimentalphysicists. The first promising results were related to the application of ML toquantum information processing; in particular, using Reinforcement Learning(RL) to control the classical part of a quantum physics problem, e.g., using anagent-based approach to control measurement-based quantum computing [5].

Although computational learning theory and quantum information process-ing have already produced good results, interdisciplinary collaboration is stillscarce in the ground. This special session tries to bring together researchersfrom both fields, with an interest in creating a collaborative research network;the session is also serving as a platform for the publication of ongoing researchin this amazing and promising field.

Some specific technical goals that are still under study and can be accom-plished by means of a collaboration between quantum information and learningtheory are the following:

1. To generalize sample and model complexity measures to the quantum case;consider mixed classical-quantum complexity measures; understand thelimits and trade-offs attainable by quantum resources and derive a quan-tum “no free lunch” theorem.

2. To establish theoretical guarantees of optimization by quantum resourceswhen applied to learning problems; analyze generalization performance;and develop operational QML algorithms.

3. To devise algorithms and protocols for blind quantum computation in re-lation to QML.

4. To characterize the role and applicability of causal Bayesian networks inML.

5. To characterize practical ML and RL tools in the optimization and controlof quantum information processing not in the asymptotic limit.

6. To devise genuinely quantum learning protocols with no classical coun-terpart, benchmarking their performance in relevant quantum informationprocessing tasks.

7. To explore quantum-like elements in symbolic systems for learning theorydevelopment.

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8. The development of models for quantum learning agents, including agentsincorporating quantum (projective) simulation methodology, and their im-plementations.

2 Quantum Machine Learning

Recent theoretical developments hint at the benefits of applying quantum meth-ods to learning algorithms [3, 4, 6, 7, 8, 9, 10]. To begin with, computationalcomplexity can be reduced exponentially in some cases, whereas quadratic re-duction can be observed in others [11]. Examples include quantum support vec-tor machines, quantum nearest neighbor clustering, quantum associative mem-ories, quantum RL [12] and several attempts to develop quantum neural net-works [7, 13, 14, 15]. On a more practical side, quantum annealing (adiabaticquantum computing) is beginning to benefit from scalable implementations, andthis hardware appears to be extremely efficient in certain global optimizationand learning problems [4].

Although improved computational complexity and reduced training time areobviously of paramount relevance, it is also important to take into account thatthrough nonconvex objective functions, QML algorithms become more robustto noise and outliers, which might make their generalization performance betterthan that achieved by many known classical algorithms. However, generalizationperformance is seldom studied in quantum algorithms, being this one of thepotential areas in which it is possible to make substantial progresses.

Another topic that deserves attention is the study of how the different typesof learning (inductive, transductive, active, supervised, unsupervised, or semi-supervised) map to quantum processes in general. One particular feature ofquantum theory is that the data encoded on a quantum state might be difficultto access by the party performing computation on them. Addressing that issuewill help generalizing known important classical results to the quantum case:sample and model complexity, the trade-off between complexity measures, “nofree lunch” theorems, and the limits attainable by using quantum resources.

From a broad perspective, the question of what learning even means in gen-eral quantum environments (e.g. when the learning agent and the environmentbecome entangled) is not trivial and needs to be investigated. Specifically, learn-ing in quantum environments does not correspond to standard quantum oraclemodels, which, in contrast, have been extensively explored. Hence, the devel-opment of new frameworks and techniques is required, in order to establish thebounds of possible quantum improvements in learning.

Many practical questions still remain to be answered: are current proposalsfor QML operational? Can they be translated to a physical implementation?For example, Grover’s search, a quantum algorithm for finding an element in anunordered set is quadratically faster than any classical version thereof, and isunderlying many QML methods, but it is hard to implement in actual quantumsystems, thus decreasing quadratic speedup due to imperfections. Quantum an-nealing suffers from similar problems: while it is possible to violate the time

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limits imposed by the gap in the adiabatic evolution and perform the process ata temperature higher than necessary, the result is likely to be a low-level excitedstate of the target model rather than the ground state. While this local optimumis still extremely useful for ML, the limits imposed by that methodology mustbe understood. By gaining insights on the trade-offs between the internal work-ing of quantum strategies and their physical implementations, further efficientquantum-inspired classical learning algorithms might be derived. Related to thisissue, the difference between thermal fluctuation and quantum fluctuation alsoneeds special attention because in many cases quantum annealing outperformssimulated annealing; therefore, it is important to know the best methods forquantum fluctuations as well as to study the possibility of improving quantumannealing by means of a sort of feedback control.

3 Bayesian networks

As stated in the abstract, an intrinsic property of QML is its embodiment ofprobability theory. This makes this field a natural relative of Statistical Machinelearning [16].

In this context, the quantum generalization of Bayesian networks and causalgraphs are of special theoretical importance, as they relate to the foundations ofquantum mechanics and nonlocality. The mathematical theory of causality, andespecially the graphical models that describe causal probabilistic relationshipshave shown to be a very useful tool in a wide range of applications, such asstatistics and ML [17]; recently, its relationship with quantum information hasalso been studied [18, 19].

In spite of its great recent success, Bayesian networks still offer many pos-sibilities for further developments. Since classical networks can be associatedwith many different types of information processing, it is important to deter-mine whether the addition of quantum effects in those problems might be atsome point determined by a Bell inequality. A straightforward possibility is thegeneralization of classical causal networks to the quantum case; that would implynot only having a long-term impact in the foundations of Physics but also provid-ing a new set of tools to understand the limitations of quantum mechanics as aresource in the processing of information. However, the latter would need a def-inition of causal networks that can entail quantum phenomena, thus developingnew techniques to perform actual learning of quantum causal networks, e.g., ex-tending variational algorithms beyond the existing state-of-the art, developingalgorithms for statistical inference of quantum systems in arbitrary quantumcausal networks and exploiting recent state-of-the art non-commutative opti-mization and factorization techniques.

4 Machine Learning and Quantum Information

While QML deals with quantum physical processes that aid learning, the use ofclassical ML algorithms to solve problems in quantum information theory is likely

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to become another cornerstone of the relationship between Physics and ML. Forthe time being, this line of research is yet to achieve wider acceptance, mostly dueto the fact that present-day learning techniques are not adapted to the nature ofquantum problems. However, there are a few promising results involving RL inadaptive quantum metrology [20]; heuristic-based RL techniques to design thequantum process tend to outperform standard greedy control techniques. Therehave also been some theoretical attemps of using RL in measurement-basedquantum computing [5], and online nonconvex optimization in circuit simula-tion [21] and ultra-cold-atom experiments [22]; next steps of those algorithmsmight deal with decoherence or noise, always present in practical applications.Apart from applying classical ML in quantum information processing, a possibleresearch avenue would entail taking ideas from learning theory to improve ordevelop new quantum strategies or protocols.

Although there are a number of proofs on how good certain quantum proto-cols and strategies can get in the asymptotic limit, even in the presence of noiseand decoherence, RL and heuristic global optimization algorithms are importantin the non-asymptotic limit, bringing these procedures closer to experimental re-ality. The key issue to explore is how close one can get to the theoretical boundsby using classical learning algorithms. Another exciting line of research is relatedto the increase of the size of the physical systems and the number of particlesthat can be achieved by adaptive quantum-enhanced metrology in a RL scenario.

The addition of computational learning theory to quantum scenarios hasalready been proven useful, but the definition of new learning schemes whereall the elements involved are of a quantum nature might not only improve thequantum information processing toolbox, but also hint at a fundamental theoryof knowledge acquisition in physical systems.

5 Machine Learning in experimental Physics

A maybe less exotic, but very useful connection between Physics and ML comesfrom the application of classsical ML to those problems in current Physics re-search in which huge amounts of data are acquired, so that there is a need togo beyond human inspection to automatically transform those raw data intostructured information and usable knowledge [23, 24].

Obviously, this is a long-term research line with a promising outlook, sincebig data appear in many problems in Physics, such as High Energy Physics forelementary-particle research, Observational Astronomy or Remote Sensing ofHyperspectral Imagery, to name a few. Therefore, there exists a need for data-based knowledge extraction procedures capable of transferring knowledge to thePhysics domain. In addition to the usual approach of using classical ML to thoseproblems, a possible and exciting future avenue of research might come from theapplication of the approaches presented in Sections 2, 3 and 4 to data acquiredin Physics problems, thus using ML approaches that are closer to Physics andwhich might provide a more realistic description of the problems.

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6 Contributions to the 24th ESANN special session

Five contributions were accepted to be part of the special session“Physics andMachine Learning: Emerging Paradigms” at ESANN 2016. The high quality ofall those contributions must be emphasized, given the low acceptance ratio ofthe conference.The accepted papers are related to different topics, most of themdescribed in previous sections, and are summarily described next.

6.1 Quantum Machine Learning

The performance of Quantum Clustering (QC) when applied to non-sphericallydistributed data sets is studied in [25]; the work shows that QC outperformsK-Means when applied to a non-spherically distributed data set. The Jaccardscore (JS) is used as performance measure; since JS can be set depending on QCparameters, local maxima can be found by tuning QC parameters, thus unveilingthe underlying data structure. The paper also suggests that a straightforwardimprovement of the approach may well be related to the discovery of a methodto obtain an appropriate number of clusters automatically. The QC algorithmintroduced in [26] was applied in this study; it uses the Schrodinger probabil-ity wave function formed as a superposition of Gaussian probability functions;looking for solutions of the harmonic oscillator potential in ground energy state,those centroids in which the potential has local minima can be found, becomingthe cluster prototypes.

A broad framework for describing learning agents in general quantum en-vironments is provided in [27]. Different classical environments that allow forquantum enhancements in learning are analyzed, by contrasting environmentsto quantum oracles. The possibility of quantum improvements depends on theinternal structure of the quantum environment; if the environments have an ap-propriate structure, there is an almost generic improvement in learning times ina wide range of scenarios.

6.2 Machine Learning and Quantum Information

In [20], authors show the relevance of well-designed classical ML algorithmsin quantum physics problems. In particular, RL is proposed to discover theoptimal sequence of actions that guarantees quantum-enhanced interferometricphase estimation up to 100 photons in a noisy environment. The study paysspecial attention to the scalability of calculations, using clustered computationand by vectorizing time-critical operations. The proposed algorithm shows tobe robust to noise.

The way of training a quantum network of pairwise interacting qubits suchthat its evolution implements a target quantum algorithm into a given networksubset is shown in [28]. The strategy followed in this work is inspired by super-vised learning and is designed to help the physical construction of a quantumcomputer operating with minimal external classical control.

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6.3 Machine Learning in experimental Physics

Authors in [29] present a ML challenge on High-Energy Physics, that was run in2014 (www.kaggle.com/c/higgs-boson) with the participation of 1785 teams.While physicists had the opportunity to improve on the state-of-the-art using“feature engineering” based on Physics principles, this was not the determiningfactor in winning the challenge. Rather, the challenge revealed that the centraldifficulty of the problem was to develop a strategy to optimize the ApproximateMedian Significance objective function directly, which is a particularly challeng-ing and novel problem. This objective function aims at increasing the power ofa statistical test. The top ranking learning machines included techniques suchas deep learning and gradient-tree boosting, two of the hottest topics in MLnowadays.

7 Acknowledgments

This work has been partially supported by the Spanish Ministry of Economyand Competitiveness under project with reference number TIN2014-52033-R.Special thanks to Dr. Peter Wittek for his guidance and for the paramount helpto set up this special session.

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